Theorem disjss2

Description: If each element of a collection is contained in a disjoint collection, the original collection is also disjoint. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Assertion	disjss2	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
3		rmoim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) → ( ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
5	4	alimdv	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( ∀ 𝑦 ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∀ 𝑦 ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
6		df-disj	⊢ ( Disj 𝑥 ∈ 𝐴 𝐶 ↔ ∀ 𝑦 ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 )
7		df-disj	⊢ ( Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑦 ∃* 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
8	5 6 7	3imtr4g	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( Disj 𝑥 ∈ 𝐴 𝐶 → Disj 𝑥 ∈ 𝐴 𝐵 ) )